Cluster categories for algebras of global dimension 2 and quivers with potential

Cluster categories for algebras of global dimension 2 and quivers with potential Claire Amiot To cite this version: Claire Amiot. Cluster categories for algebras of global dimension 2 and quivers with potential. 46 pages, small typos as it will appear in Annales de l Institut Fourier. 28. <hal-27833v2> HAL Id: hal-27833 https://hal.archives-ouvertes.fr/hal-27833v2 Submitted on 3 Jul 29 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

CLUSTER CATEGORIES FOR ALGEBRAS OF GLOBAL DIMENSION 2 AND QUIVERS WITH POTENTIAL CLAIRE AMIOT Abstract. Let k be a field and A a finite-dimensional k-algebra of global dimension 2. We construct a triangulated category C A associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When C A is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category C (Q,W) associated to a quiver with potential (Q, W). When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra J (Q, W). Contents Introduction 2 Plan of the paper 3 Acknowledgements 3 Notations 3 1. Construction of a Serre functor in a quotient category 3 1.1. Bilinear form in a quotient category 3 1.2. Non-degeneracy 4 2. Existence of a cluster-tilting object 6 2.1. t-structure on pera 7 2.2. Fundamental domain 1 3. Cluster categories for Jacobi-finite quivers with potential 17 3.1. Ginzburg dg algebra 17 3.2. Jacobian algebra 17 3.3. Jacobi-finite quiver with potentials 18 4. Cluster categories for non hereditary algebras 19 4.1. Definition and results of Keller 19 4.2. 2-Calabi-Yau property 2 4.3. Case of global dimension 2 22 5. Stable module categories as cluster categories 28 5.1. Definition and first properties 28 5.2. Case where B is hereditary 33 5.3. Relation with categories SubΛ/I w 37 5.4. Example 43 References 45 1

2 CLAIRE AMIOT Introduction The cluster category associated with a finite-dimensional hereditary algebra was introduced in [BMR + 6] (and in [CCS6] for the A n case). It serves in the representation-theoretic approach to cluster algebras introduced and studied by Fomin and Zelevinsky in a series of articles (cf. [FZ2], [FZ3], [FZ7] and [BFZ5] with Berenstein). The link between cluster algebras and cluster categories is in the spirit of categorification. Several articles (e.g. [MRZ3], [BMR + 6], [CK8], [CC6], [BMR7], [BMR8], [BMRT7], [CK6]) deal with the categorification of the cluster algebra A Q associated with an acyclic quiver Q using the cluster category C Q associated with the path algebra of the quiver Q. Another approach consists in categorifying cluster algebras by subcategories of the category of modules over a preprojective algebra associated to an acyclic quiver (cf. [GLS7a], [GLS6a], [GLS6b], [GLS7b], [BIRS7]). In both approaches the categories C (or their associated stable categories) satisfy the following fundamental properties: - C is a triangulated category; - C is 2-Calabi-Yau (2-CY for short); - there exist cluster-tilting objects. It has been shown that these properties alone imply many of the most important theorems about cluster categories, respectively stable module categories over preprojective algebras (cf. [IY6], [KR6], [KR7], [Kel8a], [Pal], [Tab7]). In particular by [IY6], in a category C with such properties it is possible to mutate the cluster-tilting objects and there exist exchange triangles. This is fundamental for categorification. Let k be a field. In this article we want to generalize the construction of the cluster category replacing the hereditary algebra kq by a finite-dimensional algebra A of finite global dimension. A candidate might be the orbit category D b (A)/ν[ 2], where ν is the Serre functor of the derived category D b (A). By [Kel5], such a category is triangulated if A is derived equivalent to an hereditary category H. However in general, it is not triangulated. Thus a more appropiate candidate is the triangulated hull C A of the orbit category D b (A)/ν[ 2]. It is defined in [Kel5] as the stabilization of a certain dg category and contains the orbit category as a full subcategory. More precisely the category C A is a quotient of a triangulated category T by a thick subcategory N which is 3-CY. This leads us to the study of such quotients in full generality. We prove that the quotient is 2-CY if the objects of T are limits of objects of N (Theorem 1.3). In particular we deduce that the cluster category C A of an algebra of finite global dimension is 2-CY if it is Hom-finite (Corollary 4.5). We study the particular case where the algebra is of global dimension 2. Since kq is a cluster-tilting object of the category C Q, the canonical candidate to be a cluster-tilting object in the category C A would be A itself. Using generalized tilting theory (cf. [Kel94]), we give another construction of the cluster category. We find a triangle equivalence C A perπ/d b Π where Π is a dg algebra in negative degrees which is bimodule 3-CY and homologically smooth. This equivalence sends the object A onto the image of the free dg module Π in the quotient. This leads us to the study of the categories perγ/d b Γ where Γ is a dg algebra with the above properties. We prove that if the zeroth cohomology of Γ is finite-dimensional, then the category perγ/d b Γ is 2-CY and the image of the free dg module Γ is a cluster-tilting object (Theorem 2.1). We show that the algebra H Γ is finite-dimensional if and only if the quotient perγ/d b Γ is Hom-finite. Thus we prove the existence of a cluster-tilting object in cluster categories C A associated with algebras of global dimension 2 which are Hom-finite (Theorem 4.1). Moreover,

CLUSTER CATEGORIES: A GENERALIZATION 3 this general approach applies to the Ginzburg dg algebras [Gin6] associated with a quiver with potential. Therefore we introduce a new class of 2-CY categories C (Q,W) endowed with a clustertilting object associated with a Jacobi-finite quiver with potential (Q, W) (Theorem 3.6). In [GLS7b], Geiss, Leclerc and Schröer construct subcategories C M of modλ (where Λ = Λ Q is a preprojective algebra of an acyclic quiver) associated with certain terminal kq-modules M. We show in the last part that the stable category of such a Frobenius category C M is triangle equivalent to a cluster category C A where A is the endomorphism algebra of a postprojective module over an hereditary algebra (Theorem 5.15). Another approach is given by Buan, Iyama, Reiten and Scott in [BIRS7]. They construct 2-Calabi-Yau triangulated categories SubΛ/I w where I w is a two-sided ideal of the preprojective algebra Λ = Λ Q associated with an element w of the Weyl group of Q. For certain elements w of the Weyl group (namely those coming from preinjective tilting modules), we construct a triangle equivalence between SubΛ/I w and a cluster category C A where A is the endomorphism algebra of a postprojective module over a concealed algebra (Theorem 5.21). Plan of the paper. The first section of this paper is devoted to the study of Serre functors in quotients of triangulated categories. In Section 2, we prove the existence of a cluster-tilting object in a 2-CY category coming from a bimodule 3-CY dg algebra. Section 3 is a direct application of these results to Ginzburg dg algebras associated with quivers with potential. In particular we give the definition of the cluster category C (Q,W) of a Jacobi-finite quiver with potential (Q, W). In section 4 we define cluster categories of algebras of finite global dimension. We apply the results of Sections 1 and 2 in subsection 4.3 to the particular case of global dimension 2. The last section links the categories introduced in [GLS7b] and in [BIRS7] with these new cluster categories C A. Acknowledgements. This article is part of my Ph. D. thesis under the supervision of Bernhard Keller. I deeply thank him for his patience and availability. I thank Bernard Leclerc, Yann Palu and Jan Schröer for interesting and helpful discussions and Idun Reiten for kindly answering my questions. I also would like to thank the referee for his interesting comments and remarks. Notations. Throughout let k be a field. By triangulated category we mean k-linear triangulated category satisfying the Krull-Schmidt property. For all triangulated categories, we will denote the shift functor by [1]. For a finite-dimensional k-algebra A we denote by moda the category of finite-dimensional right A-modules. More generally, for an additive k-category M we denote by modm the category of finitely presented functors M op modk. Let D be the usual duality Hom k (?, k). If A is a differential graded (=dg) k-algebra, we will denote by D = DA the derived category of dg A-modules and by D b A its full subcategory formed by the dg A-modules whose homology is of finite total dimension over k. We write per A for the category of perfect dg A-modules, i.e. the smallest triangulated subcategory of DA stable under taking direct summands and which contains A. 1. Construction of a Serre functor in a quotient category 1.1. Bilinear form in a quotient category. Let T be a triangulated category and N a thick subcategory of T (i.e. a triangulated subcategory stable under taking direct summands). We assume that there is an auto-equivalence ν in T such that ν(n) N. Moreover we assume that there is a non degenerate bilinear form:

4 CLAIRE AMIOT β N,X : T (N, X) T (X, νn) k which is bifunctorial in N N and X T. Construction of a bilinear form in T /N. Let X and Y be objects in T. The aim of this section is to construct a bifunctorial bilinear form: β X,Y : T /N(X, Y ) T /N(Y, νx[ 1]) k. We use the calculus of left fractions [Ver77] in the triangle quotient T /N. Let s 1 f : X Y and t 1 g : Y νx[ 1] be two morphisms in T /N. We can construct a diagram X f Y Y νx[ 1] s g t νx [ 1] s νu[ 1] νx [ 1] where the cone of s is isomorphic to the cone of s. Denote by N[1] the cone of u. It is in N since N is ν-stable. Thus we get a diagram of the form: N X u X N[1] v f Y w νx[ 1] νu[ 1] νx [ 1] νn νx, where the two horizontal rows are triangles of T. We define then β X,Y β X,Y (s 1 f, t 1 g) = β N,Y (v, w). Lemma 1.1. The form β is well-defined, bilinear and bifunctorial. as follows: Proof. It is not hard to check that β is well-defined (cf. [Ami8]). Since β is bifunctorial and bilinear, β satisfies the same properties. 1.2. Non-degeneracy. In this section, we find conditions on X and Y such that the bilinear form β XY is non-degenerate. Definition 1.2. Let X and Y be objects in T. A morphism p : N X is called a local N-cover of X relative to Y if N is in N and if it induces an exact sequence: T (X, Y ) p T (N, Y ). Let Y and Z be objects in T. A morphism i : Z N is called a local N-envelope of Z relative to Y if N is in N and if it induces an exact sequence: T (Y, Z) i T (Y, N ). Theorem 1.3. Let X and Y be objects of T. If there exists a local N-cover of X relative to Y and a local N-envelope of νx relative to Y, then the bilienar form β XY constructed in the previous section is non-degenerate.

CLUSTER CATEGORIES: A GENERALIZATION 5 For a stronger version of this theorem see also [CR]. Proof. Let f : X Y be a morphism in T whose image in T /N is in the kernel of β. We have to show that it factorizes through an object of N. Let p : N X be a local N-cover of X relative to Y, and let X be the cone of p. The morphism f is in the kernel of β, thus for each morphism g : Y νn which factorizes through νx [ 1], β(fp, g) vanishes. N p X f X N[1] Y g νx[ 1] νx [ 1] νn νx This means that the linear form β(fp,?) vanishes on the image of the morphism T (Y, νx [ 1]) T (Y, νn). This image is canonically isomorphic to the kernel of the morphism T (Y, νn) T (Y, νx). Let νi : νx νn be a local N-envelope of νx relative to Y. The sequence T (Y, νx) T (Y, νn ) is then exact. Therefore, the form β(fp,?) vanishes on Ker(T (Y, νn) T (Y, νn )). N p X X N[1] i f Y N Now β is non-degenerate on νx [ 1] νn g νx νx νi νn Coker(T (N, Y ) T (N, Y )) Ker(T (Y, νn) T (Y, νn )). Thus the morphism fp lies in Coker(T (N, Y ) T (N, Y )), that is to say that fp factorizes through ip. Since p : N X is a local N-cover of X, f factorizes through N. Proposition 1.4. Let X and Y be objects in T. If for each N in N the vector spaces T (N, X) and T (Y, N) are finite-dimensional, then the existence of a local N-cover of X relative to Y is equivalent to the existence of a local N-envelope of Y relative to X. Proof. Let g : N X be a local N-cover of X relative to Y. It induces an injection T (X, Y ) g T (N, Y ).

6 CLAIRE AMIOT The space T (N, Y ) is finite-dimensional by hypothesis. Fix a basis (f 1, f 2,...,f r ) of this space. This space is in duality with the space T (Y, νn). Let (f 1, f 2,...,f r ) be the dual basis of the basis (f 1, f 2,...,f r ). We show that the morphism Y (f 1,...,f r) r i=1 νn is a local N-envelope of Y relative to X. We have a commutative diagram: T (X, Y ) g (f 1,...,f r) T (X, νn) T (N, Y ) (f 1,...,f r) T (N, νn). If f is in the kernel of (f 1,...,f r ), then for all i = 1,..., r, the morphism f i f g is zero. Thus f g is orthogonal on the vectors of the basis f 1,...,f r and therefore vanishes. Since g is a local N-cover of X relative to Y, f is zero, and the morphism is injective. Therefore, the morphism T (X, Y ) (f 1,...,f r) T (X, νn) Y (f 1,...,f r ) r i=1 νn is a local N-envelope of Y relative to X. The proof of the converse is dual. Examples. Let A be a finite-dimensional self-injective k-algebra. Denote by T the derived category D b (moda) and by N the triangulated category pera. Since A is finite-dimensional, there is an inclusion N T. Moreover A is self-injective so of infinite global dimension. Therefore the inclusion is strict. By [KV87], there is an exact sequence of triangulated categories: pera D b (moda) moda. The derived category D b (moda) admits a Serre functor ν =? L A DA which stabilizes pera. Thus there is an induced functor in the quotient moda that we will also denote by ν. Let Σ be the suspension of the category mod A. One can easily construct (cf. [Ami8]) local N-covers and local N-envelopes, so we can apply theorem 1.3 and the functor Σ 1 ν is a Serre functor for the stable category moda. An article of G. Tabuada [Tab7] gives an example of the converse construction. Let C be an algebraic 2-Calabi-Yau category endowed with a cluster-tilting object. The author constructs a triangulated category T and a triangulated 3-Calabi-Yau subcategory N such that the quotient category T /N is triangle equivalent to C. It is possible to show (cf. [Ami8]) that the categories T and N satisfy the hypotheses of theorem 1.3. 2. Existence of a cluster-tilting object Let A be a differential graded (=dg) k-algebra. We denote by A e the dg algebra A op A. Suppose that A has the following properties: A is homologically smooth (i.e. the object A, viewed as an A e -module, is perfect); for each p >, the space H p A is zero; the space H A is finite-dimensional; g

CLUSTER CATEGORIES: A GENERALIZATION 7 A is bimodule 3-CY, i.e. there is an isomorphism in D(A e ) RHom A e(a, A e ) A[ 3]. Since A is homologically smooth, the category D b A is a subcategory of pera (see [Kel8a], lemma 4.1). We denote by π the canonical projection functor π : pera C = pera/d b A. Moreover, by the same lemma, there is a bifunctorial isomorphism DHom D (L, M) Hom D (M, L[3]) for all objects L in D b A and all objects M in pera. We call this property the CY property. The aim of this section is to show the following result: Theorem 2.1. Let A be a dg k-algebra with the above properties. The category C = pera/d b A is Hom-finite and 2-CY. Moreover, the object π(a) is a cluster-tilting object. Its endomorphism algebra is isomorphic to H A. 2.1. t-structure on pera. The main tool of the proof of theorem 2.1 is the existence of a canonical t-structure in pera. t-structure on DA. Let D be the full subcategory of D whose objects are the dg modules X such that H p X vanishes for all p >. Lemma 2.2. The subcategory D is an aisle in the sense of Keller-Vossieck [KV88]. Proof. The canonical morphism τ A A is a quasi-isomorphism of dg algebras. Thus we can assume that A p is zero for all p >. The full subcategory D is stable under X X[1] and under extensions. We claim that the inclusion D D has a right adjoint. Indeed, for each dg A-module X, the dg A-module τ X is a dg submodule of X, since A is concentrated in negative degrees. Thus τ is a well-defined functor D D. One can check easily that τ is the right adjoint of the inclusion. Proposition 2.3. Let H be the heart of the t-structure, i.e. H is the intersection D D. We have the following properties: (i) The functor H induces an equivalence from H onto ModH A. (ii) For all X and Y in H, we have an isomorphism Ext 1 H A(X, Y ) Hom D (X, Y [1]). Note that it is not true for general i that Ext i H (X, Y ) Hom D(X, Y [i]). Proof. (i) We may assume that A p = for all p >. We then have a canonical morphism A H A. The restriction along this morphism yields a functor Φ : ModH A H such that H Φ is the identity of ModH A. Thus the functor H : H ModH A is full and essentially surjective. Moreover, it is exact and an object N H vanishes if and only if H N vanishes. Thus if f : L M is a morphism of H such that H (f) =, then ImH (f) = implies that H (Imf) = and Imf =, so f =. Thus H : H ModH A is also faithful. (ii) By section 3.1.7 of [BBD82] there exists a triangle functor D b (H) D which yields for X and Y in H and for n 1 an isomorphism (remark (ii) section 3.1.17 p.85) Hom DH (X, Y [n]) Hom D (X, Y [n]). Applying this for n = 1 and using (i), we get the result.

8 CLAIRE AMIOT Hom-finiteness. Proposition 2.4. The category pera is Hom-finite. Lemma 2.5. For each p, the space H p A is finite-dimensional. Proof. By hypothesis, H p A is zero for p >. We prove by induction on n the following statement: The space H n A is finite-dimensional, and for all p n+1 the space Hom D (τ n A, M[p]) is finite-dimensional for each M in modh A. For n =, the space H A is finite-dimensional by hypothesis. Let M be in modh A. Since τ A is ismorphic to A, Hom D (τ A, M[p]) is isomorphic H (M[p]), and so is zero for p 1. Suppose that the property holds for n. Form the triangle: (H n A)[n 1] τ n 1 A τ n A (H n A)[n] Let p be an integer n+1. Applying the functor Hom D (?, M[p]) we get the long exact sequence: Hom D (τ n A, M[p]) Hom D (τ n 1 A, M[p]) Hom D ((H n A)[n 1], M[p]). By induction the space Hom D (τ n A, M[p]) is finite-dimensional. Since M[p] is in D b A we can apply the CY property. If p is n + 3, we have isomorphisms: Hom D ((H n A)[n 1], M[p]) Hom D ((H n A), M[p n + 1]) Since p n 2 is 1, this space is zero. If p = n + 2, we have the following isomorphisms. Hom D ((H n A)[n 1], M[n + 2]) DHom D (M[p n 2], H n A). Hom D ((H n A), M[3]) DHom D (M, H n A) DHom H A(M, H n A). The last isomorphism comes from lemma 2.3 (i). By induction, the space H n A is finitedimensional. Thus for p n + 2, the space Hom D ((H n A)[n 1], M[p]) is finite-dimensional. If p = n + 1 we have the following isomorphisms: Hom D ((H n A)[n 1], M[n + 1]) Hom D ((H n A), M[2]) DHom D (M, H n A[1]) DExt 1 H A(M, H n A) The last isomorphism comes from lemma 2.3 (ii). By induction, since H n A is finite-dimensional, the space Hom D ((H n A)[n 1], M[n+1]) is finite-dimensional and so is Hom D (τ n 1 A, M[n+ 1]). Now, look at the triangle τ n 2 A τ n 1 A (H n 1 A)[n + 1] (τ n 2 A)[1]. M[n + 1]

CLUSTER CATEGORIES: A GENERALIZATION 9 The spaces Hom D (τ n 2 A, M[n+1]) and Hom D ((τ n 2 A)[1], M[n+1]) vanish since M[n+1] is in D n 1. Thus we have Hom D (τ n 1 A[n 1], M[n + 1]) Hom D ((H n 1 A)[n + 1], M[n + 1]) Hom D (H n 1 A, M) Hom H A(H n 1 A, M). This holds for all finite-dimensional H A-modules M. Thus it holds for the compact cogenerator M = DH A. The space Hom H A(H n 1 A, DH A) DH n 1 A is finite-dimensional, and therefore H (n+1) A is finite-dimensional. Proof. (of proposition 2.4) For each integer p, the space Hom D (A, A[p]) H p (A) is finitedimensional by lemma 2.5. The subcategory of (pera) op pera whose objects are the pairs (X, Y ) such that Hom D (X, Y ) is finite-dimensional is stable under extensions and passage to direct factors. Restriction of the t-structure to pera. Lemma 2.6. For each X in pera, there exist integers N and M such that X belongs to D N and D M. Proof. The object A belongs to D. Moreover, since for X in DA, the space Hom D (A, X) is isomorphic to H X, the dg module A belongs to D 1. Thus the property is true for A. For the same reasons, it is true for all shifts of A. Moreover, this property is clearly stable under taking direct summands and extensions. Thus it holds for all objects of pera. This lemma implies the following result: Proposition 2.7. The t-structure on DA restricts to pera. Proof. Let X be in pera, and look at the canonical triangle: τ X X τ > X (τ X)[1]. Since pera is Hom-finite, the space H p X Hom D (A, X[p]) is finite-dimensional for all p Z and vanishes for all p by lemma 2.6. Thus the object τ > X is in D b A and so in pera. Since pera is a triangulated subcategory, it follows that τ X also lies in pera. Proposition 2.8. Let π be the projection π : pera C. Then for X and Y in pera, we have Hom C (πx, πy ) = lim Hom D (τ n X, τ n Y ) Proof. Let X and Y be in pera. An element of lim Hom D (τ n X, τ n Y ) is an equivalence class of morphisms τ n X τ n Y. Two morphisms f : τ n X τ n Y and g : τ m X τ m Y with m n are equivalent if there is a commutative square: τ n X f τ n Y τ m X g τ m Y

1 CLAIRE AMIOT where the vertical arrows are the canonical morphisms. If f is a morphism f : τ n X τ n Y, we can form the following morphism from X to Y in C: τ n X τ n Y X Y, where the morphisms τ n X X and τ n Y Y are the canonical morphisms. This is a morphism from πx to πy in C because the cone of the morphism τ n X X is in D b A. Moreover, if f : τ n X τ n Y and g : τ m X τ m Y are equivalent, there is an equivalence of diagrams: τ n X τ n Y X Y g τ m X τ m Y Thus we have a well-defined map from lim Hom D (τ n X, τ n Y ) to Hom C (πx, πy ) which is injective. Now let X be a morphism in Hom s C (πx, πy ). Let X be the cone of s. It is an X Y object of D b A, and therefore lies in D >n for some n. Thus there are no morphisms from τ n X to X and there is a factorization: We obtain an isomorphism of diagrams: τ n X X s X X X [1] X X s f f τ n X f The morphism f : τ n X Y induces a morphism f : τ n X τ n Y which lifts the given morphism. Thus the map from limhom D (τ n X, τ n Y ) to Hom C (πx, πy ) is surjective. 2.2. Fundamental domain. Let F be the following subcategory of pera: The aim of this section is to show: F = D D 2 pera. Proposition 2.9. The projection functor π : per A C induces a k-linear equivalence between F and C. Y

CLUSTER CATEGORIES: A GENERALIZATION 11 add(a)-approximation for objects of the fundamental domain. Lemma 2.1. For each object X of F, there exists a triangle with P and P 1 in add(a). Proof. For X in pera, the morphism P 1 P X P 1 [1] Hom D (A, X) Hom H (H A, H X) f H (f) is an isomorphism since Hom D (A, X) H X. Thus it is possible to find a morphism P X, with P a free dg A-module, inducing an epimorphism H P H X. Now take X in F and P X as previously and form the triangle P 1 P X P 1 [1]. Step 1: The object P 1 is in D D 1. The objects X and P are in D, so P 1 is in D 1. Moreover, since H P H X is an epimorphism, H 1 (P 1 ) vanishes and P 1 is in D. Let Y be in D 1, and look at the long exact sequence: Hom D (P, Y ) Hom D (P 1, Y ) Hom D (X[ 1], Y ). The space Hom D (X[ 1], Y ) vanishes since X is in D 2 and Y is in D 1. The object P is free, and H Y is zero, so the space Hom D (P, Y ) also vanishes. Consequently, the object P 1 is in D 1. Step 2: H P 1 is a projective H A-module. Since P 1 is in D there is a triangle τ 1 P 1 P 1 H P 1 (τ 1 P 1 )[1]. Now take an object M in the heart H, and look at the long exact sequence: Hom D ((τ 1 P 1 )[1], M[1]) Hom D (H P 1, M[1]) Hom D (P 1, M[1]). The space Hom D ((τ 1 P 1 )[1], M[1]) is zero because Hom D (D 2, D 1 ) vanishes in a t-structure. Moreover, the space Hom D (P 1, M[1]) vanishes because P 1 is in D 1. Thus Hom D (H P 1, M[1]) is zero. But this space is isomorphic to the space Ext 1 H (H P 1, M) by proposition 2.3. This proves that H P 1 is a projective H A-module. Step 3: P 1 is isomorphic to an object of add(a). As previously, since H P 1 is projective, it is possible to find an object P in add(a) and a morphism P P 1 inducing an isomorphism H P H P 1. Form the triangle Q u P v P 1 w Q[1] Since P and P 1 are in D and H (v) is surjective, the cone Q lies in D. But then w is zero since P 1 is in D 1. Thus the triangle splits, and P is isomorphic to the direct sum P 1 Q. Therefore we have a short exact sequence: H Q H P H P 1,

12 CLAIRE AMIOT and H Q vanishes. The object Q is in D 1, the triangle splits, and there is no morphism between P and D 1, so Q is the zero object. Equivalence between the shifts of F. Lemma 2.11. The functor τ 1 induces an equivalence from F to F[1] Proof. Step 1: The image of the functor τ 1 restricted to F is in F[1]. Recall that F is D D 2 pera so F[1] is D 1 D 3 pera. Let X be an object in F. By definition, τ 1 X lies in D 1 and there is a canonical triangle: τ 1 X X H X τ 1 X[1]. Now let Y be an object in D 3 and form the long exact sequence Hom D (X, Y ) Hom D (τ 1 X, Y ) Hom D ((H X)[ 1], Y ) Since X is in D 2, the space Hom D (X, Y ) vanishes. The object H X[ 1] is of finite total dimension, so by the CY property, we have an isomorphism Hom D (H X[ 1], Y ) DHom D (Y, H X[2]). But since Hom D (D 3, D 2 ) is zero, the space Hom D ((H X)[ 1], Y ) vanishes and τ 1 X lies in D 3. Step 2: The functor τ 1 : F F[1] is fully faithful. Let X and Y be two objects in F and f : τ 1 X τ 1 Y be a morphism. H X[ 1] τ 1 X X f H X H Y [ 1] τ 1 Y i Y H Y The space Hom D (H X[ 1], Y ) is isomorphic to DHom D (Y, H X[2]) by the CY property. Since Y is in D 2, this space is zero, and the composition i f factorizes through the canonical morphism τ 1 X X. Therefore, the functor τ 1 is full. Let X and Y be objects of F and f : X Y a morphism satisfying τ 1 f =. It induces a morphism of triangles: H X[ 1] τ 1 X i X f H X H Y [ 1] τ 1 Y Y H Y The composition f i vanishes, so f factorizes through H X. But by the CY property the space of morphisms Hom D (H X, Y ) is isomorphic to DHom D (Y, H X[3]) which is zero since Y is in D 2. Thus the functor τ 1 restricted to F is faithful. Step 3: The functor τ 1 : F F[1] is essentially surjective. Let X be in F[1]. By the previous lemma there exists a triangle P 1 [1] P [1] X P 1 [2]

CLUSTER CATEGORIES: A GENERALIZATION 13 with P and P 1 in add(a). Denote by ν the Nakayama functor on the projectives of modh A. Let M be the kernel of the morphism νh P 1 νh P. It lies in the heart H. Substep (i): There is an isomorphism of functors: Hom(?, X[1]) H Hom H (?, M) Let L be in H. We then have a long exact sequence: Hom D (L, P [2]) Hom D (L, X[1]) Hom D (L, P 1 [3]) Hom D (L, P [3]). The space Hom D (L, P [2]) is isomorphic to DHom D (P, L[1]) by the CY property, and vanishes because P is in D 1. Moreover, we have the following isomorphisms: Hom D (L, P 1 [3]) DHom D (P 1, L) DHom H (H P 1, L) Hom H (L, νh P 1 ). Thus Hom D (?, X[1]) H is isomorphic to the kernel of Hom H (?, νh P 1 ) Hom H (?, νh P ), which is just Hom H (?, M). Substep (ii): There is a monomorphism of functors: Ext 1 H (?, M) For L in H, look at the following long exact sequence: Hom D (?, X[2]) H. Hom D (L, P 1 [3]) Hom D (L, P 1 [3]) Hom D (L, X[2]) Hom D (L, P 1 [4]). The space Hom D (L, P 1 [4]) is isomorphic to DHom D (P 1 [1], L) which is zero since P 1 [1] is in D 1 and L is in D. Thus the functor Hom D (?, X[2]) H is isomorphic to the cokernel of Hom H (?, νh P 1 ) Hom H (?, νh P ). By defninition, Ext 1 H(?, M) is the first homology of a complex of the form: Hom H (?, νh P 1 ) Hom H (?, νh P ) Hom H (?, I), where I is an injective H A-module. Thus we get the canonical injection: Now form the following triangle: Ext 1 H (?, M) Hom D (?, X[2]) H. X Y M X[1]. Substep (iii): Y is in F and τ 1 Y is isomorphic to X. Since X and M are in D, Y is in D. Let Z be in D 2 and form the following long exact sequence: Hom D (X[1], Z) Hom D (M, Z) Hom D (Y, Z) Hom D (X, Z) Hom D (M[ 1], Z). By the CY property and the fact that Z[2] is in D, we have isomorphisms Moreover, since X is in D 3, we have Hom D (M[ 1], Z) DHom D (Z[ 2], M) Hom D (X, Z) DHom H (H 2 Z, M). Hom D (X, (H 2 Z)[2]) DHom H (H 2 Z, X[1]).

14 CLAIRE AMIOT By substep (i) the functors Hom H (?, M) and Hom D (?, X[1]) H are isomorphic. Therefore we deduce that the morphism Hom D (X, Z) Hom D (M[ 1], Z) is an isomorphism. Now look at the triangle and form the commutative diagram τ 3 Z Z H 2 Z[2] (τ 3 Z)[1] Hom D (M, τ 3 Z) Hom D (M, Z) Hom D (M, H 2 Z[2]) Hom D (M, τ 3 Z[1]) a Hom D (X[1], τ 3 Z) b Hom D (X[1], Z) c Hom D (X[1], H 2 Z[2]) d Hom D (X[1], τ 3 Z[1]) By the CY property and the fact that (τ 3 Z)[ 3] is in D, we have isomorphisms Since X is in D 3, we have Hom D (M[ 1], τ 3 Z[ 1]) DHom D (τ 3 Z[ 3], M) Hom D (X, (τ 3 Z)[ 1]) DHom H (H 3 Z, M). Hom D (X, H 3 Z[ 2]) DHom H (H 3 Z, X[1]). Now we deduce from substep (i) that a[ 1] is an isomorphism. The space Hom D (X[1], τ 3 Z[1]) is zero because X is D 3. Moreover there are isomorphisms Hom D (M, H 2 Z[2]) DHom D (H 2 Z, M[1]) DExt 1 H (H 2 Z, M). The space Hom D (X[1], H 2 Z[2]) is isomorphic to DHom D (H 2 Z, X[2]). And by substep (ii), the morphism Ext 1 H (?, M) Hom D(?, X[2]) H is injective, so c is surjective. Therefore using a weak form of the five-lemma we deduce that b is surjective. Finally, we have the following exact sequence: Hom D (X[1], Z) Hom D (M, Z) Hom D (Y, Z) Hom D (X, Z) Hom D (M[ 1], Z) Thus the space Hom D (M, Z) is zero, and Z is in D 2. It is now easy to see that there is an isomorphism of triangles: τ 1 Y Y H Y τ 1 Y [1] X Y M X[1]. Proof of proposition 2.9. Step 1: The functor π restricted to F is fully faithful. Let X and Y be objects in F. By proposition 2.3 (iii), the space Hom C (πx, πy ) is isomorphic to the direct limit lim Hom D (τ n X, τ n Y ). A morphism between X and Y in C is a diagram of the form τ n X X Y..

CLUSTER CATEGORIES: A GENERALIZATION 15 The canonical triangle yields a long exact sequence: (τ >n X)[ 1] τ n X X τ >n X Hom D (τ >n X, Y ) Hom D (X, Y ) Hom D (τ n X, Y ) Hom D ((τ >n X)[ 1], Y ) The space Hom D ((τ >n X)[ 1], Y ) is isomorphic to the space DHom D (Y, (τ >n X)[2]). The object X is in D, thus so is τ >n X, and the space DHom D (Y, (τ >n X)[2]) vanishes. For the same reasons, the space Hom D (τ >n X, Y ) vanishes. Thus there are bijections Hom D (τ n X, τ n Y ) Hom D (τ n X, Y ) Hom D (X, Y ) Therefore, the functor π : F C is fully faithful. Step 2: For X in per A, there exists an integer N and an object Y of F[ N] such that πx and πy are isomorphic in C. Let X be in pera. By lemma 2.6, there exists an integer N such that X is in D N 2. For an object Y in D N 2, the space Hom D ((τ >N X)[ 1], Y ) is isomorphic to DHom D (Y, (τ >N X)[2]) and thus vanishes. Therefore, τ N X is still in D N 2, and thus is in F[ N]. Since τ >N X is in D b A, the objects τ N X and X are isomorphic in C. Step 3: The functor π restricted to F is essentially surjective. Let X be in pera and N such that τ N X is in F[ N]. By lemma 2.11, τ 1 induces an equivalence between F and F[1]. Thus since the functor π τ 1 : pera C is isomorphic to π, there exists an object Y in F such that π(y ) and π(x) are isomorphic in C. Therefore, the functor π restricted to F is essentially surjective. Proposition 2.12. If X and Y are objects in F, there is a short exact sequence: Ext 1 D (X, Y ) Ext 1 C (X, Y ) DExt 1 D (Y, X). Proof. Let X and Y be in F. The canonical triangle yields the long exact sequence: τ < X X τ X (τ < X)[1] Hom D ((τ X)[ 1], Y [1]) Hom D (τ < X, Y [1]) Hom D (X, Y [1]) Hom D (τ X, Y [1]). The space Hom D (X[ 1], Y [1]) is zero because X is in D 2 and Y is in D. Moreover, the space Hom D (τ X, Y [1]) is zero because of the CY property. Thus this long sequence reduces to a short exact sequence: Ext 1 D (X, Y ) Hom D (τ < X, Y [1]) Hom D ((τ X)[ 1], Y [1]). Step 1: There is an isomorphism Hom D ((τ X)[ 1], Y ) DExt 1 D(Y, X). The space Hom D ((τ X)[ 1], Y [1]) is isomorphic to DHom D (Y, τ X[1]) by the CY property. Y (τ < X)[1] X[1] (τ X)[1] (τ < X)[2]

16 CLAIRE AMIOT But since Hom D (Y, (τ < X)[1]) and Hom D (Y, (τ < X)[2]) are zero, we have an isomorphism Hom D (τ X[ 1], Y ) DExt 1 D(Y, X). Step 2: There is an isomorphism Ext 1 C(πX, πy ) Hom D (τ 1 X, Y [1]). By lemma 2.11, the object τ < X belongs to F[1] and clearly Y [1] belongs to F[1]. By proposition 2.9 (applied to the shifted t-structure), the functor π : pera C induces an equivalence from F[1] to C and clearly we have π(τ < X, Y [1]) π(x). We obtain bijections Hom D (τ < X, Y [1]) Hom D (πτ < X, πy [1]) Hom D (πx, πy [1]). Proof of theorem 2.1. Step 1: The category C is Hom-finite and 2-CY. The category F is obviously Hom-finite, hence so is C by proposition 2.9. The categories T = pera and N = D b A pera satisfy the hypotheses of section 1. By [Kel8a], thanks to the CY property, there is a bifunctorial non degenerate bilinear form: β N,X : Hom D (N, X) Hom D (X, N[3]) k for N in D b A and X in pera. Thus, by section 1, there exists a bilinear bifunctorial form β X,Y : Hom C(X, Y ) Hom C (Y, X[2]) k for X and Y in C = pera/d b A. We would like to show that it is non degenerate. Since pera is Hom-finite, by theorem 1.3 and proposition 1.4, it is sufficient to show the existence of local N-envelopes. Let X and Y be objects of pera. Therefore by lemma 2.6, X is in D N. Thus there is an injection Hom D (X, Y ) Hom D (X, τ >N Y ) and Y τ >N Y is a local N-envelope relative to X. Therefore, C is 2-CY. Step 2: The object πa is a cluster-tilting object of the category C. Let A be the free dg A-module in pera. Since H 1 A is zero, the space Ext 1 D (A, A) is also zero. Thus by the short exact sequence Ext 1 D (A, A) Ext 1 C (πa, πa) DExt 1 D (A, A) of proposition 2.12, π(a) is a rigid object of C. Now let X be an object of C. By proposition 2.9, there exists an object Y in F such that πy is isomorphic to X. Now by lemma 2.1, there exists a triangle in pera P 1 P Y P 1 [1] with P 1 and P in add(a). Applying the triangle functor π we get a triangle in C: πp 1 πp X πp 1 [1] with πp 1 and πp in add(πa). If Ext 1 C (πa, X) vanishes, this triangle splits and X is a direct factor of πp. Thus, the object πa is a cluster-tilting object in the 2-CY category C.

CLUSTER CATEGORIES: A GENERALIZATION 17 3. Cluster categories for Jacobi-finite quivers with potential 3.1. Ginzburg dg algebra. Let Q be a finite quiver. For each arrow a of Q, we define the cyclic derivative with respect to a a as the unique linear map a : kq/[kq, kq] kq which takes the class of a path p to the sum p=uav vu taken over all decompositions of the path p (where u and v are possibly idempotents e i associated to a vertex i of Q). An element W of kq/[kq, kq] is called a potential on Q. It is given by a linear combination of cycles in Q. Definition 3.1 (Ginzburg). [Gin6](section 4.2) Let Q be a finite quiver and W a potential on Q. Let Q be the graded quiver with the same vertices as Q and whose arrows are the arrows of Q (of degree ), an arrow a : j i of degree 1 for each arrow a : i j of Q, a loop t i : i i of degree 2 for each vertex i of Q. The Ginzburg dg algebra Γ(Q, W) is a dg k-algebra whose underlying graded algebra is the graded path algebra k Q. Its differential is the unique linear endomorphism homogeneous of degree 1 which satisfies the Leibniz rule d(uv) = (du)v + ( 1) p udv, for all homogeneous u of degree p and all v, and takes the following values on the arrows of Q: da = for each arrow a of Q, d(a ) = a W for each arrow a of Q, d(t i ) = e i ( a [a, a ])e i for each vertex i of Q where e i is the idempotent associated to i and the sum runs over all arrows of Q. The strictly positive homology of this dg algebra clearly vanishes. Moreover B. Keller showed the following result: Theorem 3.2 (Keller). [Kel8b] Let Q be a finite quiver and W a potential on Q. Then the Ginzburg dg algebra Γ(Q, W) is homologically smooth and bimodule 3-CY. 3.2. Jacobian algebra. Definition 3.3. Let Q be a finite quiver and W a potential on Q. The Jacobian algebra J(Q, W) is the zeroth homology of the Ginzburg algebra Γ(Q, W). This is the quotient algebra kq/ a W, a Q 1 where a W, a Q 1 is the two-sided ideal generated by the a W. Remark: We follow the terminology of H. Derksen, J. Weyman and A. Zelevinsky ([DWZ7] definition 3.1). In recent works, B. Keller [Kel8b] and A. Buan, O. Iyama, I. Reiten and D. Smith [BIRS8] have shown independently the following result: Theorem 3.4 (Keller, Buan-Iyama-Reiten-Smith). Let T be a cluster-tilting object in the cluster category C Q associated to an acyclic quiver Q. Then there exists a quiver with potential (Q, W) such that End CQ (T) is isomorphic to J(Q, W).

18 CLAIRE AMIOT 3.3. Jacobi-finite quiver with potentials. The quiver with potential (Q, W) is called Jacobifinite if the Jacobian algebra J(Q, W) is finite-dimensional. Definition 3.5. Let (Q, W) be a Jacobi-finite quiver with potential. Denote by Γ the Ginzburg dg algebra Γ(Q, W). Let perγ be the thick subcategory of DΓ generated by Γ and D b Γ the full subcategory of DΓ of the dg Γ-modules whose homology is of finite total dimension. The cluster category C (Q,W) associated to (Q, W) is defined as the quotient of triangulated categories perγ/d b Γ. Combining theorem 2.1 and theorem 3.2 we get the result: Theorem 3.6. Let (Q, W) be a Jacobi-finite quiver with potential. The cluster category C (Q,W) associated to (Q, W) is Hom-finite and 2-CY. Moreover the image T of the free module Γ in the quotient perγ/d b Γ is a cluster-tilting object. Its endomorphim algebra is isomorphic to the Jacobian algebra J(Q, W). As a direct consequence of this theorem we get the corollary: Corollary 3.7. Each finite-dimensional Jacobi algebra J (Q, W) is 2-CY-tilted in the sense of I. Reiten (cf. [Rei7]), i.e. it is the endomorphism algebra of some cluster-tilting object of a 2-CY category. Definition 3.8. Let (Q, W) and (Q, W ) be two quivers with potential. A triangular extension between (Q, W) and (Q, W ) is a quiver with potential ( Q, W) where Q = Q Q ; Q 1 = Q 1 Q 1 {a i, i I}, where for each i in the finite index set I, the source of a i is in Q and the tail of a i is in Q ; W = W + W. Proposition 3.9. Denote by J F the class of Jacobi-finite quivers with potential. The class J F satisfies the properties: (1) it contains all acyclic quivers (with potential ); (2) it is stable under mutation of quivers with potential defined in [DWZ7]; (3) it is stable under triangular extensions. Proof. (1) This is obvious since the Jacobi algebra J(Q, ) is isomorphic to kq. (2) This is corollary 6.6 of [DWZ7]. (3) Let (Q, W) and (Q, W ) be two quivers with potential in J F and ( Q, W) a triangular extension. Let Q 1 = Q 1 Q 1 F be the set of arrows of Q. We have then k Q = kq R (R kf R) R kq where R is the semi-simple algebra kq and R is kq. Let W be the potential W +W associated to the triangular extension. If a is in Q 1, then a W = a W, if a is in Q 1 then a W = a W and if a is in F, then a W =. Thus we have isomorphisms J( Q, W) = k Q/ a W, a Q1 kq R (R kf R) R kq/ a W, a Q 1, b W, b Q 1 kq / b W, b Q 1 R (R kf R) R kq/ a W, a Q 1 J(Q, W ) R (R kf R) R J(Q, W). Thus if J(Q, W ) and J(Q, W) are finite-dimensional, J( Q, W) is finite-dimensional since F is finite.

CLUSTER CATEGORIES: A GENERALIZATION 19 In a recent work [KY8], B. Keller and D. Yang proved the following: Theorem 3.1 (Keller-Yang). Let (Q, W) be a Jacobi-finite quiver with potential. Assume that Q has no loops nor two-cycles. For each vertex i of Q, there is a derived equivalence DΓ(µ i (Q, W)) DΓ(Q, W), where µ i (Q, W) is the mutation of (Q, W) at the vertex i in the sense of [DWZ7]. Remark: in fact Keller and Yang proved this theorem in a more general setting. This also true if (Q, W) is not Jacobi-finite, but then there is a derived equivalence between the completions of the Ginzburg dg algebras. An other link between mutation of quivers with potential and mutations of cluster-tilting objects is given in [BIRS8] (theorem 5.1): Theorem 3.11 (Buan-Iyama-Reiten-Smith). Let C be a 2-CY triangulated category with a cluster-tilting object T. If the endomorphism algebra End C (T) is isomorphic to the Jacobian algebra J(Q, W) for some quiver with potential (Q, W), and if no 2-cycles start in the vertex i of Q, then we have an isomorphism End C (µ i (T)) J(µ i (Q, W)). Combining these two theorems with theorem 3.6, we get the corollary: Corollary 3.12. (1) If Q is an acyclic quiver, and W =, the cluster category C (Q,W) is canonically equivalent to the cluster category C Q. (2) Let Q be an acyclic quiver and T a cluster-tilting object of C Q. If (Q, W) is the quiver with potential associated with the cluster-tilted algebra End CQ (T) (cf. [Kel8b] [BIRS8]), then the cluster category C (Q,W) is triangle equivalent to the cluster category C Q. Proof. (1) The cluster category C (Q,) is a 2-CY category with a cluster-tilting object whose endomorphism algebra is isomorphic to kq. Thus by [KR7], this category is triangle equivalent to C Q. (2) In a cluster category, all cluster-tilting objects are mutation equivalent. Thus there exists a sequence of mutations which links kq to T. Moreover the quiver of a clustertilted algebra has no loops nor 2-cycles. Thus by theorem 5.1 of [BIRS8], the quiver with potential (Q, W) is mutation equivalent to (Q, ). Then the theorem of Keller and Yang [KY8] applies and we have an equivalence DΓ(Q, W) DΓ(Q, ). Thus the categories C (Q,W) and C (Q,) are triangle equivalent. By (1) we get the result. 4. Cluster categories for non hereditary algebras 4.1. Definition and results of Keller. Let A be a finite-dimensional k-algebra of finite global dimension. The category D b A admits a Serre functor ν A =? L A DA where D is the duality Hom k (?, k) over the ground field. The orbit category is defined as follows: D b A/ν A [ 2]

2 CLAIRE AMIOT the objects are the same as those of D b A; if X and Y are in D b A the space of morphisms is isomorphic to the space Hom DA (X, (νay i [ 2i]). i Z By Theorem 1 of [Kel5], this category is triangulated if A is derived equivalent to an hereditary category. This happens if A is the endomorphism algebra of a tilting module over an hereditary algebra, or if A is a canonical algebra (cf. [HR2], [Hap1]). In general it is not triangulated and we define its triangulated hull as the algebraic triangulated category C A with the following universal property: There exists an algebraic triangulated functor π : D b A C A. Let B be a dg category and X an object of D(A op B). If there exists an isomorphism in D(A op B) between DA L A X[ 2] and X, then the triangulated algebraic functor? L A X : D b A DB factorizes through π. Let B be the dg algebra A DA[ 3]. Denote by p : B A the canonical projection. It induces a triangulated functor p : D b A D b B. Let A B be the thick subcategory of D b B generated by the image of p. By Theorem 2 of [Kel5] (cf. also [Kel8c]), the triangulated hull of the orbit category D b A/ν A [ 2] is the category C A = A B /perb. We call it the cluster category of A. Note that if A is the path algebra of an acyclic quiver, there is an equivalence C Q = D b (kq)/ν [ 2] kq B /perb. 4.2. 2-Calabi-Yau property. The dg B-bimodule DB is clearly isomorphic to B[3], so it is not hard to check the following lemma: Lemma 4.1. For each X in perb and Y in D b B there is a functorial isomorphism DHom DB (X, Y ) Hom DB (Y, X[3]). So we can apply results of section 1 and construct a bilinear bifunctorial form: β XY : Hom C A (X, Y ) Hom CA (Y, X[2]) k. Theorem 4.2. Let X and Y be objects in D = D b B. If the spaces Hom D (X, Y ) and Hom D (Y, X[3]) are finite-dimensional, then the bilinear form is non-degenerate. β XY : Hom CA (X, Y ) Hom CA (Y, X[2]) k Before proving this theorem, we recall some results about inverse limits of sequences of vector spaces that we will use in the proof. Let... V p ϕ Vp 1 ϕ V 1 ϕ V be an inverse system of vector spaces (or vector space complexes) inverse system. We then have the following exact sequence V = lim V p p V p Φ q V q lim 1 V p where Φ is defined by Φ(v p ) = v p ϕ(v p ) V p V p 1 where v p is in V p. Recall two classical lemmas due to Mittag-Leffler:

CLUSTER CATEGORIES: A GENERALIZATION 21 Lemma 4.3. If, for all p, the sequence of vector spaces W i = Im(V p+i V p ) is stationary, then lim 1 V p vanishes. This happens in particular when all vector spaces V p are finite-dimensional. ϕ ϕ ϕ Lemma 4.4. Let... V p Vp 1 V 1 V be an inverse system of finitedimensional vector spaces such that V = limv p is also finite-dimensional. Let V p be the image of V in V p. The sequence V p is stationary and we have V = limv p = V. Proof. (of theorem 4.2) Let X and Y be objects of D b B such that Hom D b B(X, Y ) is finitedimensional. We will prove that there exists a local perb-cover of X relative to Y. Let P :... P n+1 P n P n 1... P be a projective resolution of X. The complex P has components in perb, and its homology vanishes in all degrees except in degree zero, where it is X. Let P n and P >n be the natural truncations, and denote by Tot(P) the total complex associated to P. For all n N, there is an exact sequence of dg B-modules: Tot(P n ) Tot(P) Tot(P >n ) The complex Tot(P) is quasi-isomorphic to X, and the complex Tot(P n ) is in perb. Moreover, Tot(P) is the colimit of Tot(P n ). Thus by definition, we have the following equalities Hom B (Tot(P), Y ) = Hom B (colim Tot(P n), Y ) = lim Hom B (Tot(P n), Y ). Denote by V p the complex Hom B (Tot(P p), Y ). In the inverse system... V p ϕ Vp 1 ϕ V 1 ϕ V, all the maps are surjective, so by lemma 4.3, there is a short exact sequence V p V p Φ q V q which induces a long exact sequence in cohomology q H 1 V q H (V ) H V p H V q. lim 1 H 1 V p limh V p We have the equalities H (V ) = H (Hom B (Tot(P), Y )) = Hom H (Tot(P), Y ) = Hom D (X, Y ). Denote by W p the complex Hom D (Tot(P p ), Y ) and by U p the complex H 1 (V p ) = Hom D (Tot(P p ), Y [ The spaces (U p ) p are finite-dimensional, so by lemma 4.3, lim 1 U p vanishes and we have an isomorphism H (lim V p ) = H (V ) lim H (V p ). The system (W p ) p satisfies the hypothesis of lemma 4.4. In fact, for each integer p, the space Hom D (Tot(P p ), Y ) is finite-dimensional because Tot(P p ) is in perb. Moreover, by the last two

22 CLAIRE AMIOT equalities W = lim W p is isomorphic to Hom D (X, Y ) which is finite-dimensional by hypothesis. By lemma 4.4, the system (W p ) p formed by the image of W in W p is stationary. More precisely, there exists an integer n such that W n = limw p. Moreover W n is a subspace of W n = Hom D (Tot(P n ), Y ) and there is an injection Hom D (X, Y ) Hom D (Tot(P n ), Y ). This yields a local perb-cover of X relative to Y. The spaces Hom D (N, X) and Hom D (X, N) are finite-dimensional for N in perb and X in D b B. Thus by proposition 1.4, there exists local perb-envelopes. Therefore theorem 1.3 applies and β is non-degenerate. Corollary 4.5. Let A be a finite-dimensional k-algebra with finite global dimension. If the cluster category C A is Hom-finite, then it is 2-CY as a triangulated category. Proof. Denote by p : D b A D b B the restriction of the projection p : B A. Let X and Y be in D b (A). By hypothesis, the vector spaces Hom D A(X, ν p b A Y [ 2p]) and Hom D A(Y, ν p b AX[ 2p + 3]) p Z are finite-dimensional. But by [Kel5], the space Hom D B(p b X, p Y ) is isomorphic to Hom D A(X, ν p b A Y [ 2p]), p so is finite-dimensional. For the same reasons, the space Hom D B(Y, X[3]) is also finite-dimensional. b Applying theorem 4.2, we get a non-degenerate bilinear form β p X,p Y. The non-degeneracy property is extension closed, so for each M and N in A B, the form β MN is non-degenerate. p Z 4.3. Case of global dimension 2. In this section we assume that A is a finite-dimensional k-algebra of global dimension 2. Criterion for Hom-finiteness. The canonical t-structure on the derived category D = D b A satisfies the property: Lemma 4.6. We have the following inclusions ν(d ) D 2 and ν 1 (D ) D 2. Moreover, the space Hom D (U, V ) vanishes for all U in D and all V in D 3. Proposition 4.7. Let X be the A-A-bimodule Ext 2 A(DA, A). The endomorphism algebra à = End CA (A) is isomorphic to the tensor algebra T A X of X over A. Proof. By definition, the endomorphism space End CA (A) is isomorphic to Hom D (A, ν p A[ 2p]) p Z For p 1, the object ν p A[ 2p] is in D 2 since νa is in D. So since A is in D, the space Hom D (A, ν p A[ 2p]) vanishes.

CLUSTER CATEGORIES: A GENERALIZATION 23 The functor ν =? L A DA admits an inverse ν 1 = L A RHom A (DA, A). Since the global dimension of A is 2, the homology of the complex RHom A (DA, A) is concentrated in degrees, 1 and 2 : H (RHom A (DA, A)) = Hom D (DA, A) H 1 (RHom A (DA, A)) = Ext 1 A (DA, A) H 2 (RHom A (DA, A)) = Ext 2 A (DA, A). Let us denote by Y the complex RHom A (DA, A)[2]. We then have Therefore we get the following equalities ν p A[2p] = A L A (Y L A p ) = Y L A p. Hom DA (A, S p A[ 2p]) = Hom DA (A, Y L A p ) = H (Y L A p ). Since H (Y ) = X, we conclude using the following easy lemma. Lemma 4.8. Let M and N be two complexes of A-modules whose homology is concentrated in negative degrees. Then there is an isomorphism H (M L A N) H (M) A H (N). Proposition 4.9. Let A be a finite-dimensional algebra of global dimension 2. The following properties are equivalent: (1) the cluster category C A is Hom-finite; (2) the functor? A Ext 2 (DA, A) is nilpotent; (3) the functor Tor A 2 (?, DA) is nilpotent; (4) there exists an integer N such that there is an inclusion Φ N (D ) D 1 where Φ is the autoequivalence ν A [ 2] of the category D = D b A and D is the right aisle of the natural t-structure of D b A. Proof. 1 2: It is obvious by proposition 4.7. 2 3 4: Let Φ be the autoequivalence ν A [ 2] of D b A. The functor Tor 2 A (?, DA) is isomorphic to H Φ and? A Ext 2 A(DA, A) is isomorphic to H Φ 1. Thus it is sufficient to check that there are isomorphisms H Φ H Φ H Φ 2 and H Φ 1 H Φ 1 H Φ 2. This is easy using Lemma 4.8 since the algebra A has global dimension 2. 4 1: Suppose that there exists some N such that Φ N (D ) is included in D 1. For each object X in C A, the class of the objects Y such that the space Hom CA (X, Y ) (resp. Hom CA (Y, X)) is finite-dimensional, is extension closed. Therefore, it is sufficient to show that for all simples S, S, and each integer n, the space Hom CA (S, S [n]) is finite-dimensional. There exists an integer p such that for all p p Φ p (S ) is in D n+1. Therefore, because of the defining properties of the t-structure, the space p p Hom D (S, Φ p (S )[n])